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On the The Aberrancy Curve
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The origin is an undulation on the curve y=px+qx2/2!+… if q = r = o. The point on the aberrancy curve corresponding to the origin is x = -3qr/P,y = 3q(3q2-Pr)/P, where P=3qs-5r2. When q = r=o, Lt.dP/dθ = 3qt-7rs = o and Lt. d2P/dθ2 = -7s2. The limiting values of x and y at the origin are respectively, given by - 3r. Lt. q/P = -3r. Lt. r/dp/dθ = -3r. Lt. S/d2p/dθ2 = 3r/7s = o, and 3(3q2-pr). Lt- q/P = 0, i.e. x = o, y = o on the aberrancy curve. So the origin also lies on the aberrancy curve.
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